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United States Patent |
5,111,483
|
Serfaty
|
May 5, 1992
|
Trellis decoder
Abstract
An improved trellis code modulation (TCM) decoder for the land mobile
environment is provided. According to the invention, a frequency
discriminator is placed at the output of the pilot extraction filter and
its output passed through a soft limiter and used to modify the metric
used by the Viterbi decoder. Under high signal conditions, the output of
the discriminator is small and the Euclidian metric is used. During deep
fades and especially during a rapid phase change, the output of the
discriminator is large and can be used to modify the metric. While the
invention is particularly useful for the land mobile environment, it can
easily be applied to other types of pilot assisted TCM schemes over
Rayleigh faded channels.
Inventors:
|
Serfaty; Salomon (Kibutz Gaash, IL)
|
Assignee:
|
Motorola, Inc. (Schaumburg, IL)
|
Appl. No.:
|
596450 |
Filed:
|
October 12, 1990 |
Current U.S. Class: |
375/341; 714/795 |
Intern'l Class: |
H04L 027/06 |
Field of Search: |
375/39,57,58,94,60
455/50,63,60
371/43
370/20
|
References Cited
U.S. Patent Documents
4945549 | Jul., 1990 | Simon et al. | 375/57.
|
Primary Examiner: Safourek; Benedict V.
Assistant Examiner: Bocure; Tesfaldet
Attorney, Agent or Firm: Egan; Wayne J.
Parent Case Text
REFERENCE TO EARLIER-FILED U.S. PATENT APPLICATION
This is a continuation-in-part of Salomon Serfaty, "Improved Trellis
Decoder," application Ser. No. 07/390,564, filed Aug. 7, 1989, now
abandoned.
Claims
What is claimed is:
1. A method for decoding a received signal comprising at least one trellis
coded modulated signal and a center-band pilot signal containing a
Rayleigh-faded component, comprising the steps of:
(a) filtering said received signal to extract said center-band pilot
signal, thus providing an extracted center-band pilot signal;
(b) applying said extracted center-band pilot signal to a frequency
discriminator to provide a frequency discriminator output;
(c) applying said frequency discriminator output to a soft limiter to
provide a metric modifier ("s");
(d) decoding said at least one trellis coded modulated signal using at
least one Viterbi decoder whose metric ("m") is modified responsive to the
formula:
m=(1-s)d.sup.2
where 1>s.gtoreq.0, and d is the Euclidian distance.
2. A radio having a decoder for decoding a received signal comprising at
least one trellis coded modulated signal and a center-band pilot signal
containing a Rayleigh-faded component, said decoder comprising:
means for filtering said received signal to extract said center-band pilot
signal, thus providing an extracted center-band pilot signal;
means for applying said extracted center-band pilot signal to a frequency
discriminator to provide a frequency discriminator output;
means for applying said frequency discriminator output to a soft limiter to
provide a metric modifier ("s");
means for decoding said at least one trellis coded modulated signal using
at least one Viterbi decoder whose metric ("m") is modified responsive to
the formula:
m=(1-s)d.sup.2
where 1>s.gtoreq.0, and d is the Euclidian distance.
3. A radio having a trellis coded quadrature amplitude modulation (QAM)
decoder for decoding a received signal comprising an upper QAM (UQAM)
signal and a lower QAM (LQAM) signal and a center-band pilot signal
containing a Rayleigh-faded component, said decoder comprising:
means for filtering said received signal to extract said center-band pilot
signal, thus providing an extracted center-band pilot signal;
means for applying said extracted center-band pilot signal to a frequency
discriminator to provide a frequency discriminator output;
means for applying said frequency discriminator output to a soft limiter to
provide a metric modifier ("s");
means for decoding said UQAM signal using a first Viterbi decoder;
means for decoding said LQAM signal using a second Viterbi decoder;
said first and second Viterbi decoders each having a metric ("m") that is
modified responsive to the formula:
m=(1-s)d.sup.2
where 1>s.gtoreq.0, and d is the Euclidian distance.
4. A method for decoding a received signal comprising a trellis coded
modulated signal and a center-band pilot signal containing a
Rayleigh-faded component, comprising the steps of:
(a) low-pass filtering said received signal to extract said center-band
pilot signal, thus providing an extracted center-band pilot signal;
(b) applying said extracted center-band pilot signal to a frequency
discriminator to provide a frequency discriminator output;
(c) applying said frequency discriminator output to a soft limiter to
provide a metric modifier ("s");
(d) decoding said trellis coded modulated signal using a Viterbi decoder
whose metric ("m") is modified responsive to the formula:
m=(1-s)d.sup.2
where 1>s.gtoreq.0, and d is the Euclidian distance.
5. A method for decoding a received signal (11) comprising a center-band
pilot signal containing a Rayleigh-faded component, an upper side band
trellis coded modulated signal and a lower side band trellis coded
modulated signal, comprising the steps of:
(a) low-pass filtering (13) said received signal to extract said
center-band pilot signal (57), thus providing an extracted center-band
pilot signal;
(b) applying said extracted center-band pilot signal (57) to a frequency
discriminator (59) to provide a frequency discriminator output;
(c) applying said frequency discriminator output to a soft limiter (65) to
provide a metric modifier ("s") (67,69);
(d) using said metric modifier s to modify the metric ("m") of a first
Viterbi decoder (53) and a second Viterbi decoder (55) responsive to the
formula;
m=(1-s)d.sup.2
where 1>s.gtoreq.0, and d is the Euclidian distance;
(e) decoding said upper side band trellis coded modulated signal using said
first Viterbi decoder (53);
(f) decoding said lower side band trellis coded modulated signal using said
second Viterbi decoder (55);
(g) combining (71) the results of step (e) and step (f) to form an output
signal (73).
6. The method of claim 1 wherein s is defined by the following equation:
##EQU1##
where the parameters A, B and C are optimized for the particular
modulation and frequency discriminator used.
7. The radio of claim 2 wherein s is defined by the following equation:
##EQU2##
where the parameters A, B and C are optimized for the particular
modulation and frequency discriminator used.
8. The radio of claim 3 wherein s is defined by the following equation:
##EQU3##
where the parameters A, B and C are optimized for the particular
modulation and frequency discriminator used.
9. The method of claim 4 wherein s is defined by the following equation:
##EQU4##
where the parameters A, B and C are optimized for the particular
modulation and frequency discriminator used.
10. The method of claim 5 wherein s is defined by the following equation:
##EQU5##
where the parameters A, B and C are optimized for the particular
modulation and frequency discriminator used.
Description
BACKGROUND OF THE INVENTION
High spectral efficient digital modulation over radio channels requires the
use of multilevel/multiphase signals. This type of signal is very
sensitive to time-varying amplitude and phase distortion associated with
land mobile communications caused by Rayleigh fading.
Prior art systems have addressed this problem by transmitting a carrier
pilot with the data modulation. This pilot is used at the receiver to
compensate for the time varying amplitude and phase distortion mentioned
above. Usually the pilot is extracted at the receiver by a bandpass
filter. Assuming there is no frequency offset between the transmitter and
the receiver, the design of the bandpass pilot extraction filter must take
into consideration the following conflicting goals:
First, its bandwidth must be wide enough to pass the spectrum associated
with the Rayleigh faded pilot signal.
Second, its bandwidth must be narrow enough to preserve the spectral
efficiency of the system, and also to keep the noise associated with the
pilot extraction as low as possible.
To improve the sensitivity of the transmission system without increasing
its bandwidth, trellis coded modulation (TCM) is used. TCM schemes use
redundant nonbinary modulation, in combination with a finite state encoder
that governs the selection of modulation signals to generate coded signal
sequences. Historically TCM has been used to improve the transmission
performance over channels where the main source of errors is additive
noise. Recently TCM has been proposed to combat the effect of Rayleigh
fading in mobile communications. However, there are problems with this
approach, as follows:
In a pilot aided transmission, the phase and amplitude of the Rayleigh
fading component could be known if the bandwidth of the pilot extraction
filter were sufficiently wide and if there were no noise associated with
the pilot extraction. However, this is unrealistic, since the limited
bandwidth of the pilot extraction filter makes that during deep fades,
when rapid phase variations are more likely to appear, the output of the
filter cannot follow the phase at its input. Also, the noise associated
with the pilot extraction is high.
Further, at the receiver, a Viterbi decoder with Euclidian metric is used
to recover the transmitted information from the trellis coded sequence.
This type of decoder is optimum if the noise associated with the samples
at the channel output is Gaussian and its samples taken at the symbol rate
are uncorrelated. However, neither assumption would apply in the case of
land mobile radio receivers.
As a result of the above, the use of a Viterbi decoder with Euclidian
metric in a Rayleigh fading environment makes performance of the TCM
scheme worse than its uncoded counterpart. The reason for this is the
rapid phase variation when the signal undergoes a deep fade. Therefore,
there is a need for an improved TCM decoder for the land mobile
environment.
SUMMARY OF THE INVENTION
Therefore, it is an object of the present invention to provide an improved
TCM decoder for the land mobile environment. Accordingly, an improved TCM
decoder is disclosed whereby a frequency discriminator is placed at the
output of the pilot extraction filter. The discriminator output is then
passed through a soft limiter whose output ("s") is used to modify the
metric used by the Viterbi decoder. Under high signal conditions, s equals
zero and the Euclidian metric is used. During deep fades and especially
during a rapid phase change, however, s is positive but less than 1, and
can be used to modify the metric in the following way:
m=(1-s)d.sup.2
where 1>s.gtoreq.0, d is the Euclidian distance, and m is the actual metric
used by the Viterbi decoder.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram that shows a first embodiment of an improved TCM
decoder according to the invention.
FIG. 2 is a graph showing the characteristics of the soft limiter of the
first embodiment.
FIG. 3 is a graph showing the performance of a TCM decoder under static
conditions.
FIG. 4 is a graph showing the performance of a TCM decoder @ 900 MHz under
Rayleigh fading conditions assuming a vehicle speed of 55 mph.
FIG. 5 is a graph showing the performance of the first embodiment under the
same conditions as in FIG. 4.
FIG. 6 is a baseband equivalent model of mobile communication system.
FIG. 7A is a frequency spectrum diagram for pilot-base transmission system
with a band edge. pilot.
FIG. 7B is a frequency spectrum diagram for pilot-based transmission system
with a band-center pilot.
FIG. 8 is a block diagram of a pilot-compensated receiver.
FIG. 9A is an RF model of a transmitter with a band-center pilot and two
QAM digital information signals.
FIG. 9B is a frequency spectrum diagram for the combined transmitted signal
of the transmitter of FIG. 9A.
FIG. 10A is a base-band equivalent model of a transmitter with a
band-center pilot and two QAM digital information signals.
FIG. 10B is a baseband equivalent frequency spectrum diagram for the
combined transmitted signal of the transmitter of FIG. 10A.
FIGS. 11A-C shaw an example of the behavior of the metric modifier
according to the invention, as a function of time.
FIG. 12 shows the input-output characteristics of the frequency
discriminator, element 59 in FIG. 1.
DETAILED DESCRIPTION OF THE INVENTION
FIG. 1 is a block diagram that shows a first embodiment of an improved TCM
decoder, according to the invention.
Referring to FIG. 1, the received signal 11 comprises a pilot and at least
one modulated signal conveying digital information. It will be appreciated
that the signal 11 may include a multiplicity of modulated signals. In
this case, the multiple modulated signals may be at frequencies above or
below the pilot, or else certain of the modulated signals may be at
frequencies above the pilot, while others may be at frequencies below the
pilot.
The modulation type may be, for example, quadrature amplitude modulation
("QAM"). Other modulation types are also possible.
The pilot signal may be, for instance, center band or band-edge.
In what follows, it will be assumed that the received signal 11 comprises a
pilot and two QAM signals, the first at 5 KHz above the pilot (herein
referred to as "UQAM"), the second at 5 KHz below the pilot (herein
referred to as "LQAM").
As shown by FIG. 1, the first step is to extract the pilot from the
composite received signal 11. This is done via a low pass filter (LPF) 13.
The output 57 in inverted (1/x) and used to multiply the incoming signal
11 via multiplier 21. This path 23 acts as a feedforward automatic gain
control. The delay 18 is necessary to off-set for the processing delay of
the LPF 13.
After processing, the output of multiplier 21 is next applied to mixers 25
and 27 via nodal point 33. As will be seen, subsequent to (or downstream
of) node 33, each individual modulated signal is separately and
independently processed by its own dedicated processing path. Since we
have assumed received signal 11 comprises two modulated signals--UQAM and
LQUAM--, then two processing paths are provided, one for each modulated
signal. Thus, the first signal, UQAM, is processed by the path consisting
of mixer 27, RX Filter 31, sampler 39B, phase adjuster 41, automatic gain
control (AGC) 45, and Viterbi Decoder 53. Similarly, the second signal,
LQAM, is processed by the path consisting of mixer 25, Filter 29, sampler
39C, phase adjuster 43, AGC 47, and Viterbi Decoder 55. It will be
appreciated that, in the event received signal 11 contained only one
modulated signal, the only one processing path downstream of node 33 would
be needed. Similarly, it will be further appreciated that, in the event
received signal contained an arbitrary number (n) of modulated signals,
then a like number (n) of processing paths would be needed.
As shown by FIG. 1, the output of multiplier 21 is next input to mixers 25
and 27 via node 33. Mixer 27 frequency shifts UQAM to a desired
intermediate frequency by a spectral amount equal to f.sub.1. Similarly,
mixer 25 frequency shifts LQAM to a desired intermediate frequency by a
spectral amount equal to f.sub.2. Assuming UQAM and LQAM are each off-set
from the desired intermediate frequency by 5 KHz, then f.sub.1 =5 KHz and
f.sub.2 =5 KHz. As a result, with the proper 5 KHz up mixing by a mixer 25
and with the proper 5 KHz down mixing by a mixer 27, and with appropriate
filtering by RX filters 29 and 31, the composite signal present at node 33
is separated into its two modulated signal components, UQAM at node 35 and
LQAM at node 37.
The UQAM component at node 35 is then sampled at the symbol rate by sampler
39B, phase-adjusted by mixer 41, and gain-corrected by AGC 45. Similarly,
the LQAM component at node 37 is sampled at the symbol rate by sampler
39C, phase-adjusted by mixer 43, and gain-corrected by AGC 47.
Synchronization and timing for these processes (sampling, phase-adjusting,
and gain-correcting) is provided by synchronization processor 52. The
processed samples 49 and 51 are then input to two Viterbi decoders 53 and
55, each one operating on one sample stream.
The output 57 of the LPF 13 is passed through a frequency discriminator 59,
sampled by sampler 39A, and delayed by delay line 61 to match the delay
through the RX filters 29 and 31 of the UQAM and LQAM components. The
input-output characteristics of frequency discriminator 59 are shown in
12. This signal 63 is then passed through a soft limiter 65 whose
input-output characteristics are depicted in FIG. 2 (see below). The s a
output 67 of soft limiter 65 is then used to modify the metric used by the
Viterbi decoder 53. Similarly, the s output 69 of soft limiter 65 is used
to modify the metric used by Viterbi decoder 55. The metric ("m") used by
each Viterbi decoder 53 and 55 is described by the following equation:
m=(1-s)d.sup.2
where d is the Euclidian distance normally used by the Viterbi decoder.
Finally, the decoded UQAM output 54 is combined with the decoded LQAM
output 56 by combiner 71, and the result provided to output 73. It will be
appreciated that, in the event the RX signal 11 contained only one
modulated signal, then this final combining step would be unnecessary.
Similarly, in the event the RX signal 11 contained an arbitrary number (n)
of modulated signals, then the combiner 71 would combine a like number (n)
of decoded outputs.
Referring now to FIG. 2, there is shown a graph showing the characteristics
of the soft limiter 65 of the first embodiment of FIG. 1. It will be
appreciated that the optimum values for the variables A,B, and C will
depend on the specific communication conditions encountered or present at
a particular time and thus will vary for each application. The optimum
values may be determined for instance, by trial and error. In one
laboratory test, the applicant determined one set of optimum values to be
as follows: A=0.022, B=0.082, and C=0.9. Results will, of course, vary.
Referring now to FIG. 3, there is shown a graph showing the performance of
a TCM decoder with a 16-QAM constellation for both UQAM and LQAM under
static conditions. The performance of the TCM schemes analyzed (four-state
and eight-state) under static conditions expressed as probability of bit
error versus E.sub.b /N.sub.o are shown. For reference, the performance of
a system using uncoded 8AM-PM in both UQAM and LQAM is also presented.
Referring now to FIG. 4, there is shown a graph showing the performance of
a four-state TCM decoder (using 16 QAM) without any metric modification at
a transmission frequency of 900 MHz under Rayleigh fading conditions
assuming a vehicle speed of 55 mph. As shown, without metric modification
and for the reasons stated above, the performance is worse than the
uncoded 8AM-PM scheme.
Referring now to FIG. 5, there is shown a graph showing performance of the
first embodiment under the same conditions as in FIG. 4. This graph shows
the performance of the two schemes under the same propagation conditions
of FIG. 4 but using now the improved TCM decoder, according to the
invention. Note the big improvement over uncoded modulation especially at
high signal-to-noise and the steep descent of the BER as the
signal-to-noise ratio is increased.
Those skilled in the art will appreciate that the improved trellis decoder,
according to the invention, may be implemented by means of an
appropriately-programmed digital signal processor ("DSP"). The DSP56000,
available from Motorola, Inc., 1301 East Algonquin Road, Schaumburg, Ill.
60196 is such a DSP, which DSP may be suitably programmed in accordance
with user's manual #DSP56000UM/AD, also available from Motorala, Inc.
The following supplemental material discusses the need and the advantage of
the modified Viterbi decoder, and is useful in understanding the
underlying theory of the invention.
To begin, further explanation may be helpful regarding the pilot signal,
and why it is used. High spectral efficiency digital modulation over radio
channels requires the use of multilevel/multiphase signals. This type of
signal is very sensitive to time varying amplitude and phase distortion
due to the Rayleigh fading inherent to this communication system. A model
of a land mobile communication system is shown in FIG. 6.
In order to overcome this problem, it is known that a pilot signal can be
sent together with the information signal. Given the RF communication
channel, the pilot can be placed anywhere in the transmission band. FIGS.
7A-B show some typical cases, i.e., a pilot at the edge of the channel
(FIG. 7A) and a pilot in the center of the channel (FIG. 7B). The band
centered pilot arrangement of FIG. 7B is preferred in some applications in
order to avoid continuous interference in an adjacent channel. At the
receiver, assuming that the channel is non-selective (flat fade), the
distortion due to the Rayleigh fading component is identical for the pilot
and for the data signal. The received pilot is used in order to compensate
for this time varying distortion in a feed-forward automatic gain control
fashion. This is expressed schematically in FIG. 8.
The upper and lower QAM signals will now be discussed. It will be
appreciated that the upper and lower QAM signals do not refer to the
in-phase and quadrature components of the quadrature amplitude modulated
signals, but rather to their relative position in the frequency domain
with respect to the pilot frequency. In the present invention, a center
frequency pilot is used, that is, the pilot is placed in the middle of the
RF transmission band. Two quadrature amplitude modulated signals are
placed to each side of the pilot. From here, the names upper QAM (UQAM)
and lower QAM (LQAM) are derived. This is more precisely seen in FIGS.
9A-B and FIGS. 10A-B. Referring now to these figures, in FIGS. 9A-B there
is shown an RF model of the transmitter used in the present invention,
while in FIGS. 10A-B there is shown a baseband equivalent model. This
particular scheme uses a band centered pilot. If a band edge pilot were to
be used, only one QAM signal would be used on one side of the pilot.
Some clarifying details about the receiver of the present invention are now
discussed. The first thing done in the receiver is a pilot extraction
process and pilot compensation, as shown in FIG. 8. The engineering
considerations leading to the design of the lowpass filter needed in the
pilot extraction process are disclosed hereinabove. If it is assumed that
the pilot correction process is ideal, then the signal information
remaining after pilot compensation is a first complex QAM signal centered
at frequency f.sub.1 and a second complex QAM signal centered at -f.sub.1,
as shown in FIG. 10B.
Referring still to FIG. 10B, it will be appreciated that at this point the
signal needs to be separated into its two QAM components (not to be
confused with the I and Q components of the individual QAM signals). This
signal processing is depicted in FIG. 1, as described hereinabove.
Referring now to FIG. 1, the signal is split in two branches, one
processing the UQAM signal and the second processing the LQAM signal. As a
result of the mixer 27, the LQAM signal, which is centered at -f.sub.1, is
translated to DC (centered around zero Hertz) and extracted by means of
the RX filter 31. This filter 31 rejects at its output the UQAM signal as
well as the pilot signal. With f.sub.2 =-f.sub.1, the same process is
repeated for the UQAM signal, using mixer 25 and RX filter 29. From this
point to the final output 73, the signal processing functions in the two
branches--including phase adjustment, AGC, sampling, and decoding via the
Viterbi decoders 53 and 55--are identical.
The metric modifier parameter `s` will now be discussed. In a communication
channel whose only impairment is additive white Gaussian noise (AWGN), the
decoding of trellis coded modulation signals is done by means of a Viterbi
decoder. In this case, it can be shown that the optimum metric to be used
by the Viterbi decoder is equal to the Euclidian distance. That is, in the
decoding process, the decoder computes the squared distance between the
received sample at the output of the channel and the points of the
transmitted constellation. Mathematically, this can be expressed as
d.sup.2 =.vertline.r.sub.n -a.sub.i .vertline..sup.2
where {a.sub.i } are the points of the complex QAM constellation. For
example, in the 16QAM system as discussed above for FIG. 3, {a.sub.i } can
have one of the following values:
{1+j1, 1+j3, 3+j1, 3+j3, -1+j1, -1+j3, -3+j1, -3+j3, 1-j1, 1-j3, 3-j1,
3-j3, -1-j1, -1-j3, -3-j1, -3-j3}.
The metric modifier operates in the following way: a frequency
discriminator centered at DC and whose sensitivity is equal to R Volts/Hz
is placed at the output of the pilot detection filter. Assume for the
moment that the channel is not Rayleigh faded, then the output of the
discriminator will be equal to zero since the only thing it sees at its
input is a DC voltage (0 Hz) corresponding to the pilot. The metric
modification scheme of the present invention stipulates that the metric to
be used by the Viterbi decoder is equal to
m=d.sup.2 (1-s)
where s is the output of the soft-limiter designated as element 65 in FIG.
1. In this case (no Rayleigh fading), it can be seen from FIG. 2 that for
an input equal to zero, the output s of the soft-limiter is also equal to
zero and therefore no modification is introduced in the metric used by the
Viterbi decoder.
In a Rayleigh fading situation, it is well known that during a deep fade,
rapid changes occur in the phase of the Rayleigh fading component. Due to
the limited bandwidth of the pilot extraction filter, those rapid changes
cannot be precisely followed at its output. Since rapid changes in the
phase correspond to large frequency excursion (frequency is equal to the
derivative of the phase), those can be detected by monitoring the output
of the discriminator. Therefore in the present invention, the
discriminator allows us to detect when a rapid change in the phase of the
Rayleigh fading component is taking place which also corresponds to a deep
fade. In such a deep fade situation, the quality of the received signal is
poor. Referring again to FIG. 2, it is seen that when the absolute value
at the output of the discriminator exceeds `B`, the output s of the soft
limiter is equal to `C`. In the example given, C=0.9, which means that, in
this case, the metric used by the Viterbi decoder is equal to
m=d.sup.2 (1-0.9)=(0.1)d.sup.2
In a very simplified but illustrative way, the Viterbi decoder is modified
to give less weight to the metric when a deep fade occurs.
It will be appreciated that the parameters A, B and C of the soft-limiter
are a function of the particular application and of R (the discriminator
sensitivity). A and B help determine when a deep fade condition occurs,
and should be a function of the sensitivity to fades of the data
transmission system under consideration. In the example discussed above
for FIG. 3, the system uses 16QAM constellations and has a transmission
frequency of 900 MHz. The Rayleigh fading corresponds to a vehicle
traveling at a speed of 55 mph. The trellis codes used have 8 and 16
states. The discriminator used has a sensitivity of 0.78 volts/KHz.
Through simulations, it was found that the best parameters for A, B and C
are those reported hereinabove, i.e., A=0.022, B=0.082 and C=0.9.
An example of the behavior of the metric modifier as a function of time for
a specific run is presented in FIGS. 11A-C. FIG. 11A shows the Rayleigh
fading component associated with the land mobile channel for a
transmission frequency of 900 MHz and for a vehicle speed of 55 mph. FIG.
11B shows the output of the discriminator. Observe in this figure that the
`spikes` at the output of the discriminator correspond to deep fades in
the channel. FIG. 11C shows the metric modifier s. It may be observed that
for a `good` signal (no fade), this metric modifier is equal to zero while
when a very deep fade occurs, it is equal to 0.9.
The beneficial effect of the technique disclosed may be clearly seen by
comparing FIGS. 4 and 5, as discussed hereinabove.
While various embodiments of the improved trellis decoder, according to the
invention, have been described herein, the scope of the invention is
defined by the following claims.
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